A (rough) pathwise approach to a class of non-linear stochastic partial differential equations

Abstract

We consider nonlinear parabolic evolution equations of the form ${\partial }_{t}u=F(t,x,Du,D^{2}u)$, subject to noise of the form $H(x,Du)\circ dB$ where $H$ is linear in $Du$ and $\circ dB$ denotes the Stratonovich differential of a multidimensional Brownian motion. Motivated by the essentially pathwise results of [Lions, P.-L. and Souganidis, P.E.; Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9] we propose the use of rough path analysis [Lyons, T. J.; Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215--310] in this context. Although the core arguments are entirely deterministic, a continuity theorem allows for various probabilistic applications (limit theorems, support, large deviations, ...).

minor changes and some more details in the appendix; this version to appear in Annales de l'Institut Henri Poincaré / Analyse non linéaire